Abstract
By an asymptotic approach previously employed for gravitational or thermocapillary motion alone, collision efficiencies are calculated for slightly deformable drops in combined gravitational and thermocapillary motion with negligible inertia and thermal convection. The constant imposed temperature gradient may be aligned with gravity in either the same or opposite direction. In the dimensionless parameter space, deformation becomes important at a smaller drop size ratio when the temperature gradient and gravity are aligned in the same direction, because the driving force is larger and induces dimple formation earlier. For the same reason, in a physical system of ethyl salicylate (ES) drops in an unbounded matrix of diethylene glycol (DEG), deformation becomes important for smaller drops when the driving forces have a parallel, rather than antiparallel, arrangement. In developing the population dynamics for slightly deformable drops, a new, simplified expression for the collision efficiency for spherical drops in the absence of van der Waals forces is presented, which successfully separates the contributions of the two driving forces. Two collisionforbidden regions can occur for opposed driving forces leading to a sharkfin shaped collision efficiency curve for two slightly deformable drops. As shown in population dynamics, if the drop distribution is broad enough, it is possible for drops to jump the first collisionforbidden region.
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Abbreviations
 A :

Hamaker constant, erg
 〈a〉:

volumeaveraged drop radius, cm
 a _{ i } :

i^{th} drop radius, μm
 b :

scaling value of dimple radius, μm
 C a :

capillary number, \(\mu _{e} V_{G,12}^{(0)}/\gamma _{0}\)
 D _{ T } :

thermal diffusivity, cm^{2}/s
 \(d_{\infty }\) :

initial horizontal offset at infinite vertical separation, cm
 E :

collision efficiency, [y_{c}/(a_{1} + a_{2})]^{2}
 f :

drop distribution function
 f _{ t } :

dimensionless tangential stress
 G :

parallel mobility function for equal but opposite external forces
 g :

gravitational constant, m/s^{2}
 h :

dimensionless drop gap, R(r − a_{1} − a_{2})/b^{2}
 J _{12} :

collision rate per unit volume, (cm^{3} ⋅ s)^{− 1}
 k :

dropsize ratio, a_{1}/a_{2}
 k _{ i } :

dispersed or external phase thermal conductivity, W/m ⋅ K
 \(\hat {k}\) :

thermal conductivity ratio, k_{d}/k_{e}
 L :

parallel gravitational mobility function
 L _{ M } :

parallel thermocapillary mobility function
 M :

transverse gravitational mobility function
 M _{ M } :

transverse thermocapillary mobility function
 N _{ F } :

modified velocity ratio
 N _{ V } :

velocity ratio, \(\pm V_{G,12}^{(0)}/V_{M,12}^{(0)}\)
 n :

number of drops per unit volume, cm^{− 3}
 p :

dimensionless pressure
 p _{12} :

pair distribution function
 Q _{12} :

interparticle force parameter
 R :

reduced drop radius, cm
 R e :

Reynolds number, \(\rho _{e} V_{G,12}^{(0)} a_{2}/\mu _{e}\)
 r :

centertocenter drop distance, cm
 s :

dimensionless centertocenter drop distance, 2r/(a_{1} + a_{2})
 T :

temperature, ^{o}C or K
 t :

dimensionless time
 t _{ i } :

timescale, s
 \(V_{12}^{(0)}\) :

relative drop velocity at infinite separation, cm/s
 y _{ c } :

critical horizontal offset at infinite vertical separation, cm
 α :

dimensionless contact force
 β :

angle between vertical and the drops’ line of centers, rad
 γ :

interfacial tension, dyn/cm
 δ :

dimensionless Hamaker parameter
 ζ :

dimensionless angular coefficient
 μ _{ i } :

dispersed or external phase viscosity, g/cm ⋅ s
 \(\hat {\mu }\) :

droptomedium viscosity ratio, μ_{d}/μ_{e}
 ξ :

dimensionless gap between drops, s − 2
 ρ _{ i } :

dispersed or external phase density, g/cm^{3}
 \(\hat \sigma \) :

dimensionless standard deviation
 ϕ :

Green’s function for axisymmetric flow
 ϕ _{0} :

volume fraction of dispersed phase
 ω :

dimensionless parameter, αζ
 0:

initial or reference value
 1:

smaller drop
 2:

larger drop
 cr :

critical
 d :

dispersed or drop phase
 e :

external or matrix phase
 G :

gravitational
 M :

Marangoniinduced or thermocapillary
 C :

combined gravitational and thermocapillary
 G :

gravitational
 T :

thermocapillary
 \({\hat {~~}}\) :

dimensionless or modified
 \(\nabla T_{\infty }\) :

applied temperature gradient, K/cm
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Acknowledgments
The authors would like to thank the Minnesota Supercomputing Institute for the use of computing resources.
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Appendices
Appendix A
The governing equations for the inner region where two drops are nearly touching, namely the thinfilm equations, are presented here in dimensionless form (Rother et al. 1997).
Normal Stress Balance
Momentum Balance
Local Boundary Integral
Mass Continuity
Integral Force Balance
Appendix B
The derivation for Eq. 13, a simplified expression for the collision efficiency \({E_{0}^{C}}\) of two spherical drops in combined gravitational and thermocapillary motion without van der Waals forces, is provided here. In the absence of deformation and attractive molecular forces, the governing equations for the critical horizontal \(y_{cr}^{C}\) for two drops interacting in the presence of gravity and a vertical temperature gradient with negligible inertia are linear, and the flow is reversible. Moreover, the velocity of the drops is linearly related to the sum of the driving forces and depends on the instantaneous position. We note here that the critical horizontal offset y_{cr} corresponds to the critical value of \(d_{\infty }\) shown in Fig. 3 demarcating the boundary between trajectories leading to collision and separation of the two drops.
In this work we consider dilute dispersions of drops such that the probability of a threebody collision is small and seek to predict the average collision rate between drops of size category 1 with those of size category 2 at a given time. As discussed elsewhere (Zhang and Davis 1991; Manga and Stone 1995), the collision rate J_{12} per unit volume, which is equal to the flux of drop pairs into the collision surface r = a_{1} + a_{2}, is given by
where p_{12}(r) is the pairdistribution function, V_{12} is the drop relative velocity, n = r/r is the outward pointing normal vector to the spherical surface r = a_{1} + a_{2}, and n_{i} is the number of drops per unit volume in size categories i = 1 or 2, characterized by radius a_{1} and a_{2}, respectively.
In the case of a dilute dispersion, the pairdistribution function must satisfy a quasisteady mass conservation equation outside the contact surface (Zhang and Davis 1991; Manga and Stone 1995):
Because the drops coalesce as they come into contact, p_{12} = 0 at r = a_{1} + a_{2}. In addition, because the drop interactions begin at wide separations, the other boundary condition is that p_{12} → 1 as \(r \to \infty \).
For the case of gravitational motion alone, thermocapillary motion alone, or combined gravitational and thermocapillary motion, the same technique can be used to find the collision rate and collision efficiency based on the symmetry of the flows. By employing (B.2) and the divergence theorem, one integrates (B.1) over the surface enclosing the volume of all trajectories starting at r = \(\infty \) and ending in contact. At infinite separation, the crosssection of this volume from symmetry considerations is a circle with radius y_{cr}. Because p_{12} = 1 and V_{12} = \(\mathbf {V}_{\mathbf {12}}^{\mathbf {(0)}}\) at r = \(\infty \), the collision rate is
where i = G, T, or C for gravitational, thermocapillary or combined motion, respectively. In the Smoluchowski limit, where there are no hydrodynamic interactions, the critical offset is y_{cr} = a_{1} + a_{2} and thus the collision efficiency for spherical drops in the absence of attractive molecular forces is
where again i = G, T or C for the appropriate motion.
Now consider the case of the collision rate for combined gravity and thermocapillarity after application of the divergence theorem:
where V_{12, C} = V_{12, G} + V_{12, T}. The volume integrals on the right side of Eqs. B.5A and B.5B are identically zero because of Eq. B.2. Evaluation of the surface integrals leads to
where the terms involving N_{V} in large parentheses are equivalent to multiplying by one and the sum is used for driving forces aligned in the same direction and the difference for opposed driving forces.
Simplifying Eq. B.6B leads to
where \(V_{12,C}^{(0)} = V_{12,G}^{(0)} \left (1 + {1 \over N_{V}} \right ) = \pm V_{12,T}^{(0)} \left (N_{V} + 1 \right )\) and the negative sign is used if N_{V} < 0.
Dividing by the collision rate in the Smoluchowski limit provides an expression for the combined collision efficiency:
Thus, Eq. 13 is recovered.
Equation B.8B is exact and does not require calculations with Eq. 2 combining mobility functions for gravitational and thermocapillary motion for each value of the relative velocity parameter N_{V}. A similar expression can also be derived for collision efficiencies for combined gravitational and thermocapillary motion of spherical drops covered with incompressible surfactant without van der Waals forces (Rother 2010) but is not presented here.
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Stark, J.K., Rother, M.A. The Combined Effects of Gravitational and Thermocapillary Driving Forces on the Interactions of Slightly Deformable, Surfactant  Free Drops. Microgravity Sci. Technol. (2020) doi:10.1007/s1221701909774y
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Keywords
 Thermocapillary
 Gravitational
 Drops
 Coalescence
 Collision Efficiency